Conductance of a STM contact on the surface of a thin film
The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film leads to a step-like dependence of differential conductance G(V) as a funct...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2012
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| Cite this: | Conductance of a STM contact on the surface of a thin film / N.V. Khotkevych, Yu.A. Kolesnichenko, J.M. van Ruitenbeek // Физика низких температур. — 2012. — Т. 38, № 6. — С. 644-652. — Бібліогр.: 32 назв. — англ. |
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Khotkevych, N.V. Kolesnichenko, Yu.A. van Ruitenbeek, J.M. 2017-05-21T16:06:22Z 2017-05-21T16:06:22Z 2012 Conductance of a STM contact on the surface of a thin film / N.V. Khotkevych, Yu.A. Kolesnichenko, J.M. van Ruitenbeek // Физика низких температур. — 2012. — Т. 38, № 6. — С. 644-652. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 74.55.+v, 85.30.Hi, 73.50.–h https://nasplib.isofts.kiev.ua/handle/123456789/117243 The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film leads to a step-like dependence of differential conductance G(V) as a function of applied bias eV. The distance between neighboring steps in eV equals the energy level spacing due to size quantization. We demonstrate that a study of G(V) for both signs of the voltage maps the spectrum of energy levels above and below Fermi surface in scanning tunneling experiments. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Электронные свойства проводящих систем Conductance of a STM contact on the surface of a thin film Article published earlier |
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Conductance of a STM contact on the surface of a thin film |
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Conductance of a STM contact on the surface of a thin film Khotkevych, N.V. Kolesnichenko, Yu.A. van Ruitenbeek, J.M. Электронные свойства проводящих систем |
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Conductance of a STM contact on the surface of a thin film |
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Conductance of a STM contact on the surface of a thin film |
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Conductance of a STM contact on the surface of a thin film |
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Conductance of a STM contact on the surface of a thin film |
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conductance of a stm contact on the surface of a thin film |
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Khotkevych, N.V. Kolesnichenko, Yu.A. van Ruitenbeek, J.M. |
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Khotkevych, N.V. Kolesnichenko, Yu.A. van Ruitenbeek, J.M. |
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Электронные свойства проводящих систем |
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Электронные свойства проводящих систем |
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2012 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
| description |
The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin
metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film
leads to a step-like dependence of differential conductance G(V) as a function of applied bias eV. The distance
between neighboring steps in eV equals the energy level spacing due to size quantization. We demonstrate that a
study of G(V) for both signs of the voltage maps the spectrum of energy levels above and below Fermi surface in
scanning tunneling experiments.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117243 |
| citation_txt |
Conductance of a STM contact on the surface of a thin film / N.V. Khotkevych, Yu.A. Kolesnichenko, J.M. van Ruitenbeek // Физика низких температур. — 2012. — Т. 38, № 6. — С. 644-652. — Бібліогр.: 32 назв. — англ. |
| work_keys_str_mv |
AT khotkevychnv conductanceofastmcontactonthesurfaceofathinfilm AT kolesnichenkoyua conductanceofastmcontactonthesurfaceofathinfilm AT vanruitenbeekjm conductanceofastmcontactonthesurfaceofathinfilm |
| first_indexed |
2025-11-25T04:24:07Z |
| last_indexed |
2025-11-25T04:24:07Z |
| _version_ |
1850503617285455872 |
| fulltext |
© N.V. Khotkevych, Yu.A. Kolesnichenko, and J.M. van Ruitenbeek, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6, pp. 644–652
Conductance of a STM contact on the surface of a thin film
N.V. Khotkevych and Yu.A. Kolesnichenko
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: khotkevych@ilt.kharkov.ua
J.M. van Ruitenbeek
Kamerlingh Onnes Laboratorium, Universiteit Leiden, Postbus 9504, 2300 Leiden, The Netherlands
Received January 12, 2012
The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin
metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film
leads to a step-like dependence of differential conductance G(V) as a function of applied bias eV. The distance
between neighboring steps in eV equals the energy level spacing due to size quantization. We demonstrate that a
study of G(V) for both signs of the voltage maps the spectrum of energy levels above and below Fermi surface in
scanning tunneling experiments.
PACS: 74.55.+v Tunneling phenomena: single particle tunneling and STM;
85.30.Hi Surface barrier, boundary, and point-contact devices;
73.50.–h Electronic transport phenomena in thin films.
Keywords: STM, surface barrier, thin film, conductance.
1. Introduction
Today a fairly large number of papers have addressed
the problem of calculating point-contact conductance for
use in analyzing and interpreting scanning tunneling mi-
croscopy (STM) experiments (for reviews see, for exam-
ple, [1,2]). The low symmetry of the problem and the wide
variety of objects under study do not allow developing a
general theory of STM, and different approaches for spe-
cific problems are used. The theory papers on this subject
can be divided into two groups: One uses methods taking
into account the specific atomic structure of the STM tip
and that of the test specimen. These methods make it poss-
ible to reproduce the crystallographic structure of the sam-
ple surface in the calculated STM images and this is very
useful for arriving at a correct interpretation of experimen-
tal data. The main deficiency of this approach is the lack of
analytical formulas for the STM current–voltage characte-
ristics as numerical calculations must be performed for
every specific case. The other group of works exploit sim-
plified models of noninteracting electrons which allows
finding relatively simple analytical expressions that de-
scribe the STM current qualitatively. For this reason such
theoretical results are widely used by experimentalists.
One of the first free-electron models describing STM
experiments was proposed by Tersoff and Hamann [3]
whose theoretical analysis of tunnel current is based on
Bardeen’s formalism [4], in which a tunneling matrix ele-
ment is expressed by means of independent wave functions
for the tip and the sample within the barrier region. Using
the model wave functions the authors [3] showed that the
conductance of the system is proportional to the local den-
sity of states of the sample at the tip position. In principle it
is possible to extract information on subsurface objects
(single defects, clusters, interfaces, etc.) by STM, but this
requires a more detailed theoretical analysis [5], which
takes into account the influence of subsurface electron
scattering on the tunneling current.
The physical picture of an electron tunneling through a
classically forbidden region is that the electron flow
emerging from the barrier is defined by the matching of the
wave functions of carriers incident on the barrier and those
that are transmitted. For a three-dimensional STM geome-
try the wave functions for electrons transmitted through the
vacuum region radically differs from the electron wave
functions in an isolated sample and they describe the elec-
tron propagation into the bulk from a small region on the
surface below the STM tip. In contrast, the theory of Ref. 3
and its modifications (see [1,2] and references therein) uses
unperturbed wave functions of the surface Bloch states.
Changes in the wave functions of transmitted electrons due
to scattering by subsurface objects provide the information
about such scattering in the STM conductance.
Conductance of a STM contact on the surface of a thin film
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6 645
In Ref. 6 it was proposed to introduce in the theory of
STM the model by Kulik et al. [7]. In this model a three-di-
mensional STM tip is replaced by an inhomogeneous barrier
in an otherwise nonconducting interface that separates the
two conductors. In Ref. 7 it was shown that under assump-
tion of small transparency of the tunnel barrier the wave
function (and thus the current–voltage characteristics) can be
found analytically for an arbitrary size of the tunnel area.
The results in [6] for the conductance of the tunnel point
contact were generalized to an arbitrary Fermi surface for
the charge carriers in Refs. 8, 9. In a series of papers the
model [7] has been expanded to describe oscillations of the
STM conductance resulting from electron scattering by sub-
surface defects [6,8–11] (for reviews see [12]).
Scanning tunneling microscopes have been widely used
for the study of various small-sized objects: islands, thin
films deposited on bulk substrates, etc. [13–20]. First, a dis-
crete periodic spatial variation of the STM current originat-
ing from the quantization of electron states was observed in
the quantum wedge: a nanoscale flat-top Pb island on a
stepped Si(111) surface [13]. Later these authors showed
that the lattice structure of an interface buried under a film
of Pb, whose thickness can be as many as 10 times the Fer-
mi wavelength, can be clearly imaged with STM [14]. They
concluded that the key to the transparency of a metal lies in
a highly anisotropic motion of the electrons and the strong
quantization of their transverse wave function components.
In the paper [15] the electronic states of thin Ag films grown
on GaAs(110) surfaces was investigated by STM with sin-
gle-layer thickness resolution, and the quantum-well states
arising from the confinement geometry of the Ag films have
been identified. Quantum size effects, manifested in the
formation of new electronic bound states, were investigated
by STM on thin Pb islands of varying heights on the
Si(111)-(7×7) surface in Ref. 16. In experiments [17] it was
demonstrated that scanning tunneling microscopy and
spectroscopy of epitaxial Pb islands on Si(111) reveal
adiabatic lateral modulation of the energy spectra of the
quantum well, providing remote electronic images of the
subsurface reflection phase. In Ref. 18 a step structure at
the buried Pb on Si(111) 6×6-Au interface was determined
by utilizing the presence of quantum well states. It was
demonstrated that the spatial step positions as well as the
step heights can be extracted nondestructively and with
atomic layer precision by STM. Vertical Friedel oscilla-
tions in interface-induced surface charge modulations of
Pb islands of a few atomic layers on the incommensurate
Si(111)–Pb surface have been observed [19]. Thus, de-
tailed experimental results have been obtained, but a mi-
croscopic theory for STM tunneling spectra on samples of
finite size has not been reported, which provides the moti-
vation for the present work. Current–voltage characteristics
for size quantization in planar thin film geometries of met-
al–insulator–metal tunneling junctions have been investi-
gated theoretically in Refs. 21, 22. Standing electron wave
states in thin Pb films have been observed by electron
tunneling in early experiments by Lutskii et al. [23].
In this paper we present the differential conductance
(G V ) for small contacts, having a radius a smaller than
the Fermi wave length = / ,F Fpλ where Fp is the
Fermi momentum. The contacts are formed on the surface
of a thin metal film and we analyze the voltage dependence
of ( )I V and ( ).G V We focus on the size quantization ef-
fects of the electron energy spectrum in the film on ( ).G V
The organization of this paper is as follows. The model
that we use to describe the contact, and the method for
obtaining a solution of the three-dimensional Schrödinger
equation asymptotic in the small radius of the contact, are
described in Sec. 2. In Sec. 3 the current–voltage characte-
ristics and the differential conductance are found on the
basis of a calculation of the probability current density
through the contact. Section 4 presents a physical interpre-
tation of the results obtained. In Sec. 5 we conclude by
discussing the possibilities for exploiting these theoretical
results for interpretation of electron energy spectroscopy in
thin films by STM. In the Appendixes we solve the
Schrödinger equation for the tunnel point contact in
framework of our model (Appendix 1) and for a point con-
tact without barrier (Appendix 2) and find the wave func-
tions for electrons transmitted through the contact. These
solutions are used in Sec. 3 for the calculation of current.
2. Model and electron wave function of the system
The model that we consider is illustrated in Fig. 1. Elec-
trons can tunnel through an orifice centered at the point
= 0r in an infinitely thin insulating interface at = 0z
from a conducting half-space (the tip) into a conducting
sheet of thickness d (Fig. 1,b). The radius a of the contact
and the thickness d of the film are assumed to be much
smaller than the shortest mean free path, i.e., we consider a
purely ballistic problem. The wave function ψ satisfies
the Schrödinger equation
[ ]2
2
2( ) ( ) ( ) = 0.m U
∗
∇ ψ + ε − ψr r r (1)
In Eq. (1) m∗ and ε are electron effective mass and ener-
gy, respectively. The inhomogeneous potential barrier in
the plane = 0z we describe by the function ( ) =U r
0 ( ) ( ),U f z= δρ where = ( , )x yρ is a two-dimensional posi-
tion vector in the plane and 1( ) = ( ),f a− Θ −ρρ with ( )xΘ
the Heaviside step function. For such model the wave func-
tion ( )ψ r satisfies the following boundary conditions at the
interface = 0z and at the metal sheet surface =z d
( , 0) = ( , 0),ψ + ψ −ρ ρ (2)
0
2
2
( , 0) ( , 0) = ( ) ( ,0),z z
m U
f
∗
′ ′ψ + −ψ − ψρ ρ ρ ρ (3)
( , ) = 0.dψ ρ (4)
N.V. Khotkevych, Yu.A. Kolesnichenko, and J.M. van Ruitenbeek
646 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6
Equations (1)–(4) can be solved in the limit of a small con-
tact, 1ka ( = 2 /k m∗ε is the absolute value of the
electron wave vector ).k In the zeroth approximation in
the contact diameter the solutions of Eq. (1) for 0z ≷ are
independent and satisfy the zero boundary condition
( ,0) = 0ψ ρ at the impenetrable interface at = 0.z The
quantum states in the conducting half-space ( < 0)z (the
tip) are defined by the three components of the electron
wave vector = ( , ),zkk k with k a two-dimensional vec-
tor parallel to the interface. In the metal film (0 < )z d<
the quantum states are characterized by a two-dimensional
vector κ perpendicular to the z axis and by the discrete
quantum number n ( = 1, 2, ...)n which results from the
finite size of the conductor in the z direction. The energy
eigenvalues and eigenfunctions for the two disconnected
conductors are given by
2 2 2 2 2( )
= ,
2 2
zk k k
m m∗ ∗
+
ε ≡ (5)
||
0 ( ) = 2 e sin , < 0,i
zi k z zψ kr ρ (6)
and
2 2 2( )
= , = 1, 2,...,
2
znk
n
m∗
κ +
ε (7)
0 ( ) = 2 e sin , 0 < < ,i
zni k z z dψ −r κρ (8)
where = / .znk n dπ In Eqs. (6) and (8) we use a wave
function normalization with unit amplitude of the wave
incident to the interface.
The partial wave for the first order approximation 1( )ψ r
in the small parameter 1,ka which describes the transi-
tion of electrons from one to the other conductor, is given in
the Appendixes. Appendix 1, Eqs. (A1.5) and (A1.6), gives
the solution for a tunnel point contact, having a potential
barrier of small transparency 2
0= / 1t k m U∗ at the
orifice in the plane = 0.z Appendix 2, Eqs. (A2.6)–(A2.8),
gives the solutions for a contact without barrier. Figure 2
illustrates the spacial variation of the square modulus of the
wave function for electrons transmitted through the contact
into the film.
3. Current–voltage characteristic and conductance of a
point contact
As has been shown in Ref. 24 for a ballistic point contact
of small radius ,a with a much smaller than the electron
mean free path ,l the electrical potential ( )V r drops over a
Fig. 1. Schematic representation of a STM experiment on a thin metal film (a) and the model that we employ to represent the contact
between a bulk conductor (tip) and a metallic film (b). The dashed picture of the tip in a illustrates a metallic point contact (STM tip
touches the surface). Electron trajectories in b are shown schematically.
Isulation
substratez
ba
Metal film
Isulation substrate
d
Area of
tunnelingArea of
tunneling
Metal film
Tip Tip
2a
2a
2a
Tip
d
V
V z = V( )
V z( ) = 0
V = 0
Fig. 2. Space distribution of the square modulus of the wave
function for electrons injected by a STM tip into a metal sheet of
thickness = 15,d where = 2 / kλ π is the electron wave length,
= / 2 .λ π
Conductance of a STM contact on the surface of a thin film
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6 647
distance r a∼ from the contact, and in the limit 0a → the
potential ( )V r can be approximated by a step function
( ).V zΘ − In this approximation, for the calculation of the
electrical current we can take the electron distribution func-
tions ( )f ∓ at 0z ≶ as the Fermi functions Ff with ener-
gies shifted by the applied bias eV (e is the negative electron
charge) , ( ) = ( ( )).Ff f eV zε − Θ −∓ Figure 3 illustrates the
occupied energy states in the two conductors for both signs
of the applied bias .eV At > 0eV the electrons flow from
the bulk conductor (the tip) into the film and, vice-versa, at
< 0eV they flow from the film into the massive conductor.
The total current through the area of the contact can be found
by integration over the flux ( )J ± in both directions
( )
=1
1( ) = ( )(1 ( ))
2 F F
n
I V d J f f eV
d
∞ ∞
−
−∞
ε − ε − −
π ∑∫ κ
( )
3
2 ( )(1 ( )).
(2 )
F Fd J f eV f
∞
+
−∞
− ε − − ε
π ∫ k (9)
In Eq. (9) we integrate over the wave vector k in the semi-
infinite conductor for the current in the negative direction
(second term), and integrate over the two-dimensional
wave vector κ and sum over the discrete quantum number
n for the opposite direction of the current (first term).
For simplicity we will take the temperature to be zero.
In this case the electric current is defined by electrons
passing the contact in one direction only, depending on the
sign of the applied bias. The flux ( )J ± integrated over the
area of the contact is calculated in the usual way
2
( )
1 1
= 00 0
= Im ( , ) ( , ) ,
a
z
e
J d d z z
zm
π
± ∗
∗
±
∂⎡ ⎤ρρ ϕ ψ ψ⎢ ⎥∂⎣ ⎦∫ ∫ ρ ρ (10)
where ( cos , sin ).ρ ϕ ρ ϕ=ρ The wave function 1( , )zψ ρ
should be taken as the wave transmitted through the con-
tact, given by Eqs. (A1.5) and (A2.6) with = zkk for elec-
tron flux from the tip to the sheet, ( ) ,J + and by Eqs.
(A1.6) and (A2.7) with = znkk ( = 1, 2 ...)n for fluxes
( )J − in the opposite direction. The energy shift eV in the
region < 0z should be taken into account, which for our
choice of the reference point of energy (see Fig. 3) implies
that the absolute value of the electron wave vector in the
half-space < 0z is given by = 2 ( ) / .k m eV∗ ε −
For the tunnel point contact (tpc) the flux can be ex-
pressed in terms of the wave function in the contact plane
(A1.1), and we obtain
24 4 5 2
( )
3 3 2
0
cos
( 1)(2 1),
12
tpc
e a k
J N N N
m d U
+
∗
π ϑ
+ + (11)
and
34 5 2
( )
3 2
0
.
6
zn
tpc
e a k k
J
m U
−
∗
π
− (12)
Here ϑ is the angle between the vector k and the z axis,
and ( ) = [ / ]N k kd π with [ ]x the integer part of .x
For a metallic point contact (mpc) without barrier the
expressions for the flux ( )
mpcJ ± are written by means of
Eqs. (A2.4), (A2.9)–(A2.11),
22 6 2
( )
3
cos
( 1)(2 1),
9
mpc
e a k
J N N N
m d
+
∗
π ϑ
+ + (13)
and
36 2
( ) 8
.
9
zn
mpc
e a k k
J
m
−
∗
π
− (14)
Fig. 3. Illustration of the occupied energy states at zero temperature in the two conductors for both signs of the applied bias :eV
> 0eV (a), < 0eV (b).
� �
eV > –� �N F+1
|eV| > �N�F –
�N +1
�F
�F + eV
�N
eV
z
0 d
�F
�F + eV
�N
�N –1
z0 d
a b
N.V. Khotkevych, Yu.A. Kolesnichenko, and J.M. van Ruitenbeek
648 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6
Substituting Eqs. (11)–(14) into the general expression (9)
we find the current–voltage characteristic of the system
2
20
23 5 2
2( ) = ( ) ( ), 0
( )
F
F
k
F Fk
I dkk meVI V k S k eV
k d k
−∫
(15)
and
2
0
23
1 2 1( ) = ( )
5 3 3( )
F
F FF
eV eVI
I V S k
k d
⎧ ⎡ ⎤⎛ ⎞⎪ ⎢ ⎥− + + +⎜ ⎟⎨ ⎢ ⎥ε ε⎝ ⎠⎪ ⎣ ⎦⎩
2
3 3 5 5
2( ) ( ) ( ) ( )
3
F FF F
F F
eV
S k S k S k S k
k d
⎛ ⎞π⎡ ⎤ ⎡ ⎤+ − + − ×⎜ ⎟⎣ ⎦ ⎣ ⎦ε⎝ ⎠
3 5
7 7
F
1 ( ) ( )
5
F F
F F
eV
S k S k
k d k d
⎛ ⎞ ⎛ ⎞π π⎡ ⎤× + − −⎜ ⎟ ⎜ ⎟⎣ ⎦ε⎝ ⎠ ⎝ ⎠
2
2
4 8( ) 1 , 0,
5 3 3
F
F
F F F
eV eVkS k eV
k
⎡ ⎤⎛ ⎞⎢ ⎥− + − ⎜ ⎟⎢ ⎥ε ε⎝ ⎠⎣ ⎦
(16)
where 2 2= / 2F Fk m∗ε is the Fermi energy,
2 4 5 8
0, 3 2
0
= ,
12
F
tpc
e a k
I
m U∗
π
(17)
6 8
0, = ,
9
F
mpc
e a k
I
m∗
(18)
( )mS k is a finite sum of mth powers of integer numbers
( )
=1
( ) = .
N k
m
m
n
S k n∑ (19)
Note that ( ) ( ),m mS k H N−≡ where ( )mH n are generalized
harmonic numbers. The current is plotted in Fig. 4 as a func-
tion of bias voltage for two choices of the film thickness.
Differentiating Eqs. (15) and (16) with respect to voltage we
obtain the differential conductance ( ) = /G V dI dV for a
point contact with radius ,Fa λ
2
1 2 23
1( ) = ( ) ( ) , 0;
2
F
F
k
F
F
F F k
kG V G S k dkk S k eV
k k
⎧ ⎫
⎪ ⎪−⎨ ⎬
⎪ ⎪⎩ ⎭
∫
(20)
1 2 3 3
4( ) = 1 ( ) 4 ( ) ( )
3
FF F
F F F
eV eV
G V G S k S k S k
k d
⎧ ⎡ ⎤ π⎪ ⎡ ⎤+ + − +⎨ ⎢ ⎥ ⎣ ⎦ε ε⎪ ⎣ ⎦⎩
3
5 5
4 ( ) ( )
3
F F
F
S k S k
k d
⎛ ⎞π ⎡ ⎤+ − −⎜ ⎟ ⎣ ⎦⎝ ⎠
2
21 4 8 ( ) , 0.
3
F F
F F F
eV eVk S k eV
k
⎫⎡ ⎤⎛ ⎞ ⎪⎢ ⎥− + − ⎜ ⎟ ⎬⎢ ⎥ε ε⎝ ⎠ ⎪⎣ ⎦ ⎭
(21)
In the limit 0eV → the zero-bias conductance taken from
both sides coincides, as it should,
1
1 2(0) = ( ) = ( 1)(2 1),
6F F F F
G
G G S k N N N+ + (22)
where = ( ),F FN N k and 1G is the conductance of the
contact between the bulk conductor (the tip) and a thin film
that has only a single energy level available below Fε for
the motion along z,
3
1 0 3
3= (0) .
( )F
G G
k d
π (23)
0 (0)G is the conductance of a contact between two con-
ducting unbound half-spaces. For a tunnel point contact
this is given by [7,12]
( ) ( )
2 422
0,
0
0 = ,
36
FF
tpc
e k ak
G
m U∗
⎛ ⎞
⎜ ⎟⎜ ⎟ π⎝ ⎠
(24)
and for a metallic point contact we have [25]
2 6
0, 3
8 ( )
(0) = .
27
F
mpc
e k a
G
π
(25)
For d →∞ Eqs. (20) and (21) transform into the known
voltage dependence of the conductance for a point contact
between unbound conducting half-spaces [27],
3
0 0
1( ) = (0) 1 .
3F F
eV eV
G V G
⎡ ⎤⎛ ⎞⎢ ⎥+ − ⎜ ⎟⎢ ⎥ε ε⎝ ⎠⎣ ⎦
(26)
The dependence of the differential conductance ( )G V
for both signs of applied voltage is illustrated in Fig. 5. For
comparison the dependence 0 0( ) / (0)G V G from Eq. (26)
is also shown.
Fig. 4. Dependence of the total current, ( ),I V on the applied bias
over the point contact for two choices of the thicknesses of the
metal film. The constant 0I is given by Eq. (17) or Eq. (18).
–1.0 –0.5 –0.5 –1.0
0
–2
2
4
6
8
eV/�F
k dF = 10
k dF = 100
I I/ 0
Conductance of a STM contact on the surface of a thin film
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6 649
4. Discussion
Thus, in the framework of the model illustrated in
Fig. 1 we have obtained the current–voltage characteristic
and the differential conductance for a contact on the sur-
face of a thin metal film. Under the assumption that the
contact radius a is much smaller than the Fermi wave
length Fλ we found asymptotically exact formulas for the
dependence of the total current ( )I V (Eqs. (15) and (16))
and the contact conductance ( )G V (Eqs. (20), (21)) on the
applied voltage. In the limit of zero temperature and neg-
lecting scattering processes we have demonstrated that the
( )I V dependence has kinks and ( )G V undergoes jumps at
the same values of applied bias eV (see Figs. 4 and 5).
These events result from the size quantization of the elec-
tron spectrum in the film.
The results obtained show that even in Ohm’s-law ap-
proximation (22), 0,eV → the conductance ( )G V is not
simply proportional to the electron density of states (DOS)
in the isolated film,
2( ) = .F
f
m N
d
∗
ρ ε
π
(27)
It is remarkable that the dependence of the conductance
(0)G (22) on the number of quantum levels FN is the
same for, both, tunnel and metallic point contacts. This fact
shows that such dependence is not sensitive to the model
taken for the potential barrier, and that it is the result of the
point-contact geometry. Recently, the relationship between
the differential conductance and the local density of states
has been studied in a tight-binding approximation for tun-
nel junctions, where the junction geometry can be varied
between the limiting cases of a point-contact and a planar
junction [28]. In the framework of a real-space Keldysh
formalism the authors of Ref. 28 have shown that the diffe-
rential conductance is not, in general, proportional to the
sample DOS for planar junctions, although features of the
DOS may be present.
From Eqs. (20) and (21) it follows that the conductance
is nonsymmetric in the applied bias. This asymmetry can
be explained as follows: Let > 0eV and electrons tunnel
from the bulk conductor into the film (Fig. 3,a) in which
FN subbands of the size quantization are partially filled.
If the bias eV is smaller than the distance Δε between the
Fermi level Fε and the bottom of the next (empty) sub-
band 2 2 2 2
1 = ( 1) / 2 ,N FN m d∗+ε π + 1= ,N F+Δε ε − ε
the electron can tunnel into any of the FN subbands. At
=eV Δε tunneling into the ( 1)-thFN + subband becomes
possible and the conductance ( )G V undergoes a positive
jump. Such jumps are repeated for increasing voltage for
all higher subbands. For < 0,eV when electrons tunnel
from the thin film into bulk metal (Fig. 3,b) the situation is
somewhat different. If the bias eV becomes larger than
distance Δε between the bottom of the last partially filled
subband 2 2 2 2= / 2N FN m d∗ε π and Fermi energy,
= ,F NΔε ε − ε the contribution of the -thFN subband to
the tunnel current does not depend on the voltage because
for any >eV Δε all electrons of this subband can tunnel
into the bulk states of the left conductor. For this reason
the differential conductance drops for values of eV coin-
ciding with bottoms of subbands of size quantization in the
film. The distance between neighboring jumps of the con-
ductance on the voltage scale equals the distance between
energy levels 2 2 2
1= = (2 1) / 2 .N N N FN m d∗+Δε ε − ε π +
For < 0eV the number of conductance jumps is finite and
equals the number of discrete levels below Fermi surface
.FN The asymmetry around = 0V and the general shape
of the jumps in the conductance can be recognized in the
experiments, see, e.g., [15]. In the special case of a 2D
electron system, which has only one level in the potential
well, there is a single negative jump of G(V). Such a jump
has been observed in Ref. 29 by STM investigations of the
2D electron gas at noble-metal surfaces. For > 0eV the
number of conductance jumps formally is not restricted.
However, for > FeV ε our approach is no longer applica-
ble and the influence of field emission on the tunnel cur-
rent must be taken into account [30,31].
The observation of manifestations of the size quantiza-
tion in the STM conductance requires a few conditions
which must be fulfilled: The distance between the energy
levels must be large enough and should satisfy the condi-
tion / , ,N TΔε τ where τ is the mean scattering time
of the electrons in the film and T is the temperature. The
surfaces of the metal film in the region of the contact must
be atomically smooth [32]. When a finite lifetime of the
quantized states becomes relevant, the temperature broa-
dening of the Fermi function, or surface imperfections
need to be taken into account this will result in a rounding
of the jumps in the curve ( )G V presented in Fig. 5
(Eqs. (20), (21)), which was plotted under assumptions of
Fig. 5. Dependence of the normalized differential conductance,
0( ) / (0),G V G on the applied bias over the point contact for two
choices of the thicknesses of the metal film. The voltage depen-
dence for a point contact between two semi-infinite bulk conduc-
tors is shown for comparison (short-dashed curve).
–1.0 0 1.0
1.0
1.5
2.0
2.5
eV/�F
k dF = 10
k dF = 100
conducting half-spaces
G
V
G
(
)/
(0
)
0
N.V. Khotkevych, Yu.A. Kolesnichenko, and J.M. van Ruitenbeek
650 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6
perfectly specular surfaces, = 0,T and .τ → ∞ With these
restrictions taken into account the current–voltage curves
in Fig. 4 give a fair qualitative description of the experi-
mental results of Ref. 14.
It can be easily seen that the results obtained have a
more wide domain of applicability than that of a rectangu-
lar well for the conducting film. For any model of the po-
tential which restricts the electron motion in one direction
the differential conductance has a step-like dependence on
the applied bias with distances between the steps equal to
the distances between the quantum levels.
5. Conclusion
In summary, we have investigated the conductance of
ultra small contacts, for which the radius is smaller than
the Fermi wave length, on top of the surface of a thin film.
The discreteness of the component of the electron momen-
tum transverse to the film surface is taken into account,
where the distance between the electron energy levels due
to the size quantization is assumed to be larger than the
temperature. Both, a contact with a potential barrier of low
transparency, and a contact without barrier have been con-
sidered. In framework of our model, using a δ-function
potential barrier, the current–voltage characteristic ( )I V
of the system and differential conductance ( )G V have
been obtained. We predict a sawtooth dependence of
( )G V on the applied bias and show that the distance be-
tween neighboring jumps is equal to the distance between
neighboring energy levels of size quantization, i.e., this
dependence can be used for spectroscopy of size quantized
levels. At > 0eV the jumps in the conductance are posi-
tive and correspond to distances between levels above the
Fermi surface, while ( )G V undergoes negative jumps for
< 0,eV the distances between which are equal to the dis-
tances between the levels below the Fermi surface. The
predicted quantization of the conductance can be observed
in STS experiments, and the shape of the theoretical curves
agrees well with experiments.
Appendix 1: Electron tunneling between the tip
and the thin film
We search a solution to Eq. (1) at = 0V in the form of
a sum 0 1=ψ ψ +ψ for the incident and backscattered
waves, and 1=ψ ψ for the transmitted wave. Here 0ψ , as
given by Eqs. (6), (8), is the unperturbed wave function
that does not depend on the barrier amplitude 0 ,U while
1 01/ Uψ ∼ gives the first order correction. Substituting
the wave function into the boundary conditions (2) and (3)
one should match terms of the same order in 01/ .U As a
result the boundary condition (3) becomes [7]
2
1
0
( ,0) = e ( ),ii a
m U∗
ψ − Θ −ρ
k κρρ (A1.1)
where = zkk when the wave is incident to the contact
from the tip side, and = znkk when the wave arrives at the
contact from the sheet. For 1ka we have in the plane of
the contact 1κρ and we can neglect the exponent in the
boundary condition (A1.1).
The function 1( , )zψ ρ can be represented as a Fourier
integral
1( , ) = e ( , ).iz d z
∞
′−
−∞
′ ′ψ Ψ∫ κ ρρ κ κ (A1.2)
The Fourier components in (A1.2) should satisfy the ze-
ro boundary condition at = ,z d but are otherwise freely
propagating along z,
sin ( )
( , ) = ( ,0) , 0 ,
sin
z
z
k z d
z z d
k d
′ −′ ′Ψ Ψ
′
κ κ (A1.3)
( , ) = ( ,0) exp ( ), 0,zz ik z z′ ′ ′Ψ Ψ −κ κ (A1.4)
with 2 2= ,zk k′ ′− κ = 2 / .k m∗ε From Eqs. (A1.1),
(A1.2) it follows that
2
1
2
0
( )1( ,0) = e ( ,0) = .
(2 ) 2
i J ai ad
m U
∞
′
∗
−∞
′κ′Ψ ψ −
′κπ π∫
kκ ρκ ρ ρ
(A1.5)
Substituting this into Eq. (A1.2) we find the wave func-
tions for the electrons transmitted through the contact as
2
1 0 1
0 0
sin ( )
( , ) = ( ) ( ) ,
sin
z
z
k d zi az d J J a
k dm U
∞
∗
′ −′ ′ ′ψ κ κ ρ κ
′∫
k
ρ
0 < ,z d (A1.6)
2
1 0 1
0 0
( , ) = ( ) ( ) exp ( ),z
i az d J J a ik z
m U
∞
∗
′ ′ ′ ′ψ κ κ ρ κ −∫
k
ρ < 0,z
(A1.7)
where ( )nJ x is the Bessel function of the first kind.
Appendix 2: Metallic point contact between STM tip
and metal film
Here we consider a point contact without potential bar-
rier in the plane of the interface. When the contact radius is
small, 1,ka we can use perturbation theory for the elec-
tron wave function in the limit 0.a → In zeroth approxi-
mation the wave functions are given by Eqs. (6) and (8).
The first order correction, 1( ,0),ψ ρ to the wave function in
the plane of the contact can be found by the method pro-
posed in [25]. For distances r λ from the contact we
can neglect the second term in the Schrödinger equation
(1) and it reduces to the Laplace equation. We express the
wave function in coordinates of an oblate ellipsoid of revo-
lution ( , , ),σ τ ϕ with 0σ and 1 1.− τ As a conse-
quence of the cylindrical symmetry of the problem the
Conductance of a STM contact on the surface of a thin film
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 6 651
wave function 1( , )ψ σ τ does not depend on .ϕ The inter-
face corresponds to = 0τ and the plane of the orifice is at
= 0.σ In these coordinates we obtain the equation
2 21 1(1 ) (1 ) = 0,
∂ψ ∂ψ∂ ∂⎡ ⎤ ⎡ ⎤+ σ + − τ⎢ ⎥ ⎢ ⎥∂σ ∂σ ∂τ ∂τ⎣ ⎦ ⎣ ⎦
(A2.1)
with the boundary condition at the interface
1( > 0, = 0) = 0.ψ σ τ (A2.2)
The solution of the boundary problem (A2.1), (A2.2) is
1 1 2( , ) = [ (1 arctan )],c cψ σ τ τ σ + + σ σ (A2.3)
where 1c and 2c are constants. For = 0σ Eq. (A2.3)
gives the function 1( , )zψ ρ in the plane of the contact
= 0,z aρ
2
1 2 2( ,0) = 1 .c
a
ρ
ψ −ρ (A2.4)
As in Appendix 1, we express 1( , )zψ ρ as a Fourier
integral and, using the Eq. (A2.4), we find for the Fourier
components
1
1 22
( )1( ,0) = e ( ,0) = ,
(2 )
i j a
d c a
∞
′
−∞
′κ′Ψ ψ
′κπ ∫ κ ρκ ρ ρ (A2.5)
where 1( )j x is the spherical Bessel function of the first
kind. Substituting Eq. (A2.5) into Eq. (A1.2) and using
Eqs. (A1.3), (A1.4) we obtain
2
1 0 1
0
sin ( )
( , ) = ( ) ( ) ,
2 sin
z
z
c a k d z
z d J j a
k d
∞ ′ −′ ′ ′ψ ρ κ κ ρ κ
π ′∫
0 < ,z d (A2.6)
and
2
1 0 1
0
( , ) = ( ) ( )e ,
2
ikzc a zz d J j a
∞
′−′ ′ ′ψ ρ κ κ ρ κ
π ∫ < 0.z
(A2.7)
The constant 2c must be found from the boundary condi-
tion (3) at 0 = 0,U which for this case takes the form
1 1( , 0) ( , 0)
2 = 0.i
z z
∂ψ ρ + ∂ψ ρ −
− −
∂ ∂
k (A2.8)
The meaning of the symbol k is explained below
Eq. (A1.1). Differentiating Eqs. (A2.6) and (A2.7) with
respect to z and calculating the integrals in the limit of
small a we find
3
1 2
2 3
= 0
( 1)(2 1) ,
2 2 18z
c a ai N N N
z a d+
⎡ ⎤∂ψ π π
− + + +⎢ ⎥
∂ π ⎢ ⎥⎣ ⎦
(A2.9)
3
1 2
2
= 0
,
2 92z
c a k ai
z a−
⎛ ⎞∂ψ π π
+⎜ ⎟⎜ ⎟∂ π ⎝ ⎠
(A2.10)
where = [ / ]N kd π with [ ]x the integer part of .x Substi-
tuting Eqs. (A2.9) and (A2.10) into (A2.8) in leading ap-
proximation in ,a in which only first terms in brackets
(proportional to 21/ )a should be taken into account, we
find for the unknown constant
2 2 .c i ak (A2.11)
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